samples/texts/2274792/page_7.md

C.2 Uniparental Teaching with Transmission Errors

This model is identical to the uniparental teaching model in the main text except that teaching has an error rate, $\epsilon$, which is the frequency with which a mother pays a teaching cost, but her offspring fails to learn the behavior. Equation 2 is replaced by Equation C.6 and Equation 3 is replaced by Equation C.7.

$$x_{sk1}^{\mathrm{\prime }\mathrm{\prime }}=\frac{1}{2}(\underset{\text{Allele from mother}}{\underbrace{({\tau }_{sk}-ϵ)x_{fk1}^{\mathrm{\prime }}}}+\underset{\text{Allele from father}}{\underbrace{(x_{mk0}^{\mathrm{\prime }}+x_{mk1}^{\mathrm{\prime }})\sum _h({\tau }_{sh}-ϵ)x_{fh1}^{\mathrm{\prime }}}})(C.6)x\text{'}\text{'}\_\{sk1\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash underbrace\{(\tau \_\{sk\}\; -\; ϵ)x\text{'}\_\{fk1\}\}\_\{\backslash text\{Allele\; from\; mother\}\}\; +\; \backslash underbrace\{(x\text{'}\_\{mk0\}\; +\; x\text{'}\_\{mk1\})\; \backslash sum\_h\; (\tau \_\{sh\}\; -\; ϵ)x\text{'}\_\{fh1\}\}\_\{\backslash text\{Allele\; from\; father\}\}\; \backslash right)\; \backslash quad\; (C.6)$$

$$x_{sk0}^{\mathrm{\prime }\mathrm{\prime }}=\frac{1}{2}(\underset{\text{Allele from mother}}{\underbrace{x_{fk0}^{\mathrm{\prime }}+(1-{\tau }_{sk}+ϵ)x_{fk1}^{\mathrm{\prime }}}}+\underset{\text{Allele from father}}{\underbrace{(x_{mk0}^{\mathrm{\prime }}+x_{mk1}^{\mathrm{\prime }})\sum _h(x_{fh0}^{\mathrm{\prime }}+(1-{\tau }_{sh}+ϵ)x_{fh1}^{\mathrm{\prime }})}})(C.7)x\text{'}\text{'}\_\{sk0\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash underbrace\{x\text{'}\_\{fk0\}\; +\; (1\; -\; \tau \_\{sk\}\; +\; ϵ)x\text{'}\_\{fk1\}\}\_\{\backslash text\{Allele\; from\; mother\}\}\; +\; \backslash underbrace\{(x\text{'}\_\{mk0\}\; +\; x\text{'}\_\{mk1\})\; \backslash sum\_h\; (x\text{'}\_\{fh0\}\; +\; (1\; -\; \tau \_\{sh\}\; +\; ϵ)x\text{'}\_\{fh1\})\}\_\{\backslash text\{Allele\; from\; father\}\}\; \backslash right)\; \backslash quad\; (C.7)$$

As in the main text, I ran numeric simulations of this system for various combinations of teaching costs, t, and reproductive benefits of the cultural trait, b, and an innovation rate of r = 0.005. Since more than 90% of females learn the trait from their mother I conservatively set ϵ to 0.1 as the error rate for the simulation. As in the main text, for each parameter combination, I ran the simulation starting with four initial allele frequencies. In each frequency one allele started at 5% of the population and the rest were evenly distributed among the rest of the population. I started the frequency of the cultural trait at zero in all four initial conditions and ran the simulations until they converged to a shared equilibrium or ran for 10⁷ generations.